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June  2022, 15(3): 385-401. doi: 10.3934/krm.2021034

Glassey-Strauss representation of Vlasov-Maxwell systems in a Half Space

Yunbai Cao 1, and  Chanwoo Kim 2,3,

1. 

Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA
2. 

Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53717, USA
3. 

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

This paper is dedicated to the memory of the late Bob Glassey

Received  June 2021 Published  June 2022 Early access  November 2021

Fund Project: CK is supported in part by National Science Foundation under Grant No. 1900923 and the Brain Pool program (NRF-2021H1D3A2A01039047) of the Ministry of Science and ICT in Korea

Following closely the classical works [5]-[7] by Glassey, Strauss, and Schaeffer, we present a version of the Glassey-Strauss representation for the Vlasov-Maxwell systems in a 3D half space when the boundary is the perfect conductor.

Keywords: Vlasov-Maxwell system, Glassey-Strauss representation, half space, perfect conductor, electromagnetic fields.
Mathematics Subject Classification: Primary: 35Q61, 35Q83; Secondary: 35Q70.
Citation: Yunbai Cao, Chanwoo Kim. Glassey-Strauss representation of Vlasov-Maxwell systems in a Half Space. Kinetic and Related Models, 2022, 15 (3) : 385-401. doi: 10.3934/krm.2021034
References:
[1]

Y. Cao and C. Kim, On some recent progress in the Vlasov-Poisson-Boltzmann system with diffuse reflection boundary, Recent Advances in Kinetic Equations and Applications, Springer INdAM Series, 48 (2021).

[2]

Y. Cao, C. Kim and D. Lee, Global strong solutions of the Vlasov-Poisson-Boltzmann system in bounded domains, Arch. Rational Mech. Anal., 233 (2019), 1027-1130.  doi: 10.1007/s00205-019-01374-9.

[3]

J. Cooper and W. Strauss, The initial boundary problem for the Maxwell equations in the presence of a moving body, SIAM J. Math. Anal., 16 (1985), 1165-1179.  doi: 10.1137/0516086.

[4]

G. Federici, et al, Plasma-material interactions in current tokamaks and their implications for next step fusion reactors, Nucl. Fusion, 41 (2001).

[5]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics, 1996. doi: 10.1137/1.9781611971477.

[6]

R. T. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions. Ⅰ, Arch. Rational Mech. Anal., 141 (1998), 331-354.  doi: 10.1007/s002050050079.

[7]

R. T. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions. Ⅱ, Arch. Rational Mech. Anal., 141 (1998), 355-374.  doi: 10.1007/s002050050080.

[8]

R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., 92 (1986), 59-90.  doi: 10.1007/BF00250732.

[9]

R. T. Glassey and W. A. Strauss, High velocity particles in a collisionless plasma, Math. Methods Appl. Sci., 9 (1987), 46-52.  doi: 10.1002/mma.1670090105.

[10] D. Griffith, Introduction to Electrodynamics, Cambridge University Press, 4th edition, 2017. 
[11]

Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions, Comm. Math. Phys., 154 (1993), 245-263.  doi: 10.1007/BF02096997.

[12]

Y. Guo, Regularity for the Vlasov equations in a half space, Indiana Univ. Math. J., 43 (1994), 255-320.  doi: 10.1512/iumj.1994.43.43013.

[13]

J. D. Jackson, Classical Electrodynamics, Wiley, 3rd edition, 1998. doi: 10.1119/1.19136.

[14]

X. Liu, F. Yang, M. Li and S. Xu, Generalized boundary conditions in surface electromagnetics: Fundamental theorems and surface characterizations, Appl. Sci., 9 (2019). doi: 10.3390/app9091891.

[15]

J. Luk and R. M. Strain, A new continuation criterion for the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 331 (2014), 1005-1027.  doi: 10.1007/s00220-014-2108-8.

[16]

K. McDonald, Electromagnetic fields inside a perfect conductor, Unpublished note.

[17]

T. T. Nguyen, T. V. Nguyen and W. A. Strauss, Global magnetic confinement for the 1.5D Vlasov-Maxwell system, Kinetic and Related Models, 8 (2015), 153-168.  doi: 10.3934/krm.2015.8.153.

show all references

References:
[1]

Y. Cao and C. Kim, On some recent progress in the Vlasov-Poisson-Boltzmann system with diffuse reflection boundary, Recent Advances in Kinetic Equations and Applications, Springer INdAM Series, 48 (2021).

[2]

Y. Cao, C. Kim and D. Lee, Global strong solutions of the Vlasov-Poisson-Boltzmann system in bounded domains, Arch. Rational Mech. Anal., 233 (2019), 1027-1130.  doi: 10.1007/s00205-019-01374-9.

[3]

J. Cooper and W. Strauss, The initial boundary problem for the Maxwell equations in the presence of a moving body, SIAM J. Math. Anal., 16 (1985), 1165-1179.  doi: 10.1137/0516086.

[4]

G. Federici, et al, Plasma-material interactions in current tokamaks and their implications for next step fusion reactors, Nucl. Fusion, 41 (2001).

[5]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics, 1996. doi: 10.1137/1.9781611971477.

[6]

R. T. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions. Ⅰ, Arch. Rational Mech. Anal., 141 (1998), 331-354.  doi: 10.1007/s002050050079.

[7]

R. T. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions. Ⅱ, Arch. Rational Mech. Anal., 141 (1998), 355-374.  doi: 10.1007/s002050050080.

[8]

R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., 92 (1986), 59-90.  doi: 10.1007/BF00250732.

[9]

R. T. Glassey and W. A. Strauss, High velocity particles in a collisionless plasma, Math. Methods Appl. Sci., 9 (1987), 46-52.  doi: 10.1002/mma.1670090105.

[10] D. Griffith, Introduction to Electrodynamics, Cambridge University Press, 4th edition, 2017. 
[11]

Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions, Comm. Math. Phys., 154 (1993), 245-263.  doi: 10.1007/BF02096997.

[12]

Y. Guo, Regularity for the Vlasov equations in a half space, Indiana Univ. Math. J., 43 (1994), 255-320.  doi: 10.1512/iumj.1994.43.43013.

[13]

J. D. Jackson, Classical Electrodynamics, Wiley, 3rd edition, 1998. doi: 10.1119/1.19136.

[14]

X. Liu, F. Yang, M. Li and S. Xu, Generalized boundary conditions in surface electromagnetics: Fundamental theorems and surface characterizations, Appl. Sci., 9 (2019). doi: 10.3390/app9091891.

[15]

J. Luk and R. M. Strain, A new continuation criterion for the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 331 (2014), 1005-1027.  doi: 10.1007/s00220-014-2108-8.

[16]

K. McDonald, Electromagnetic fields inside a perfect conductor, Unpublished note.

[17]

T. T. Nguyen, T. V. Nguyen and W. A. Strauss, Global magnetic confinement for the 1.5D Vlasov-Maxwell system, Kinetic and Related Models, 8 (2015), 153-168.  doi: 10.3934/krm.2015.8.153.

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